Dynamics of Dual Scale-Free Polymer Networks.

Mircea Galiceanu,Oliver Mülken,Maxim Dolgushev,Luan Tota De Carvalho

doi:10.3390/polym9110577

Mircea Galiceanu, Oliver Mülken + Show 2 more

Open Access

https://doi.org/10.3390/polym9110577

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Abstract

We focus on macromolecules which are modeled as sequentially growing dual scale-free networks. The dual networks are built by replacing star-like units of the primal treelike scale-free networks through rings, which are then transformed in a small-world manner up to the complete graphs. In this respect, the parameter describing the degree distribution in the primal treelike scale-free networks regulates the size of the dual units. The transition towards the networks of complete graphs is controlled by the probability p of adding a link between non-neighboring nodes of the same initial ring. The relaxation dynamics of the polymer networks is studied in the framework of generalized Gaussian structures by using the full eigenvalue spectrum of the Laplacian matrix. The dynamical quantities on which we focus here are the averaged monomer displacement and the mechanical relaxation moduli. For several intermediate values of the parameters’ set , we encounter for these dynamical properties regions of constant in-between slope.

Highlights

Nowadays, in different areas of science, such as physics, chemistry, biology, economics, the study of complex networks becomes of huge significance
Inspired by recent experimental techniques allowing chemical transformations to be made from hyperbranched polymers to functional core–shell nanogel systems [10], and due to our interest in the fundamental role of the presence of loops in polymer networks, we study in this article a new kind of polymer network—the dual scale-free networks
The limiting topologies that one can get as a function of this power-law exponent, γ, are networks made of huge dual units for very low values of γ and linear chains for very high values of γ

Summary

Introduction

In different areas of science, such as physics, chemistry, biology, economics, the study of complex networks becomes of huge significance. We construct dual scale-free polymers by using the procedure implemented in reference [15]. The dual units considered in this article range from rings to complete graphs The transition between these units is implemented by adding, with probability p, links between nodes from the same ring. In this way, we get the ring limit for p = 0 and for p = 1 we obtain complete graphs. The relaxation dynamics of polymers is completely determined by knowing all eigenvalues and eigenvectors of the connectivity (Laplacian) matrix, which allows one to study very large systems. The paper is structured as follows: In Section 2 we briefly describe the algorithm used to construct the dual scale-free networks.

Sequentially Growing Dual Scale-Free Networks

Theoretical Model

Eigenvalues Spectrum

Relaxation Dynamics

Conclusions